By Ralph Abraham, Christopher D. Shaw
Dynamics: The Geometry of habit (Studies in Nonlinearity)
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Extra info for Dynamics: The Geometry of Behavior (Studies in Nonlinearity)
As it gets closer, it slows down. It gets closer and slower indefinitely and approaches the critical point asymptotically. That is, it takes forever to reach C. We say that C is the limit point of the trajectory through A. This trajectory approaches its limit point asymptotically. 3. These are the time series corresponding to the three trajectories of the preceding phase portrait. The graph (time series) of the trajectory of C is a horizontal line, a constant function of time. This represents the constant trajectory of the critical point.
These models succeed remarkably well and have been used by many satisfied customers over the years. Some of these examples are presented in the next four chapters. But some obstinate readers may still exclaim: SO WHAT- Well, dynamical systems theory has yet more to offer: PREDICTION FOREVER. Sophisticated techniques from the research frontier of pure mathematics have been employed to yield qualitative predictions of the asymptotic behavior of the system in the long run, or even forever. Although qualitative predictions are not as precise as quantitative ones, they are a whole lot better than no predictions at all.
If its limit set is an attractor, it belongs to a basin. So, if it belongs to the separatrix (and therefore not to a basin), it must tend to a nonattractor. Thus, the separatrix consists of insets of exceptional limit sets. The preceding examples are artificial, made up to illustrate the concepts. But we are overdue for some more meaningful examples. So, at this point, let's turn to gradient systems-a rich source of simple examples, based on a geometrical construction. 6. Gradient Systems The gradient operation of vector calculus provides dynamical systems (vectorfields) of an especially simple type called a gradient system.